# Integration of log x

What is Integration of log x. Friends, today we are going to talk about the integration of log x, so let’s start.

solution of Integration in log x

\int~log~x ~dx

So now let’s integration log x

\int~log~x~dx~
=\int~(log~x)~.1~dx~

Using by parts

Rule of ILATE

1st function = f(x) = log x & 2nd function = g(x) = 1

Now we know that

Formula of by parts

\int~f(x)~.~g(x)~dx = f(x)~\int~g(x)~dx~-~\int \big( \acute{f(x)~\int~g(x)dx}\big) dx

Putting value of f(x) and g(x)

\int~log~x~.~1~dx = log~x~\int~1~dx~-~\int \big( \acute{\frac{d(log~x)}{dx}~\int~1.dx}\big) dx
=(log~x).x~-~\int~\frac{1}{x}.x~dx

=x~log~x~-~\int~1.~dx
x~log~x~-~x~+C\\
or\\
x(log~x~-~1)~+C`

## The answer for Integration of log x is, x log x – x + c , or x (log x – 1 ) + C

Definition of Logarithm

## what are logarithms

The logarithm of a positive number whose base is any other positive number other than the unit is the exponent of that power , which, if placed on the base, becomes equal to the required number, is called logarithm.

Example :- The logarithm of 10000000 (one crore) in base 10 will be 7 because by adding 7 to the base 10 its value becomes 10000000.

That is, for a number x, base b and exponent n,

If a, x, N are three numbers such that a^x = N (a > 0, a 1) then the exponent x is said to be the logarithm of N on base a.

∴ x = logₐ N

#### Special circumstances :-

(i). If there is a finite quantity by adding base zero , then the logarithm of 1 is always equal to zero.

∵ a⁰ = 1,
∴ logₐ 1 = 0

(ii). The logarithm of a number whose base is the same number is equal to 1.

∵ a¹ = a,
∴ logₐ a = 1

## logarithm formula

• logₐ(m × n) = logₐ m + logₐ n
• logₐ m/n = logₐ m – logₐ n
• logₐ mⁿ = n logₐ m
• logₐ a = 1/logₐ b
• logₐ a = log a/log b
• log₍ₐⁿ₎ N = n.logₐ N [if a>0]
• loge m = 2.3026 log₁₀ m
• log₁₀ m = 0.4343 logₑ m

## how to find logarithm formula

Rule 1. The logarithm of the multiplication of any two numbers x and y is equal to the sum of the logarithms of these two numbers.

There is a condition in this rule that the base of both these numbers must be the same, only then this theorem is proved.

• logₐ (x × y) = logₐ x + logₐ y

Rule 2. The division of two numbers is equal to the logarithm of the difference of their logarithms.

There is a condition in this rule that both the numbers must have the same base.

• log x/y = log x – log y

Rule 3. The logarithm of one number to another base can be determined from any base of the same number.

• loga x = logb x × loga b
• logb x = loga x / loga b

Rule 4 . The logarithm of a number raised to a certain degree is equal to the number obtained by multiplying the logarithm of that number by the index of the power . Both these numbers have the same base.

• logb xn = n logb x

## parts of logarithm

The logarithm of any number has two parts.

• Integer
• Mantissa

### 1. Integer

The integer is the whole part of the logarithm . It can be positive or negative.

• positive integer
• negative integer

(a). Positive Integer:- If the value of a number is more than one, then the integer of its logarithm is positive and its value is one less than the number of digits to the left of the decimal in that number.

#### Examples of positive integers:-

• The full moon of 4321 will be 3.
• The full term of 432.1 will be 2.
• The integer of 43.21 will be 1.
• The integer of 4.321 will be 0.

(b). Negative Integer:- If the value of a number is less than one, then the integer of its logarithm is negative and its value is one more than the number of zeros to the right of the decimal in the number.

#### Examples of negative integers:

• The integer of 0.6212 is -1 and it is written as 1 (bar one).
• The integer of 0.04212 is +2 and it is written as 2 (bar two).

### 2. Mantissa

The value of the mantissa is always positive. The decimal part of the logarithm of a number is called the mantissa.

Let the logarithms of any two numbers be 2.3010 and 2.4771 respectively. The integer 2 and the mantissa of the logarithm are 0:3010 and both are positive. The integers of the second logarithm are negative. But the value of its mantissa (0.4771) is positive.

## logarithm questions

Q.1 Which of the following would be false?
A. log₁₀₀ 1 = 0
B. logₑ 1 = 0
C. logₐ 1 = 1
D. log¹⁰ = 1

Ans. logₐ 1 = 1 is false.

Q.2 log x/log a = ?
A. logₑ a
B. logₐ x
C. x/a
D. x – a

Ans. log e/log a = logₐ x

Q.3 What is log m + log n equal to?
A. logₑ (m/n)
B. logₑ (m × n)
C. logₑ (mⁿ)
D. 1

Ans. log m + log n = logₑ (m × n)

Q.4 What is the value of log₂ 64 equal to?
A. 2
B. 4
C. 6
D. 8

log₂ 64
log₂ (2)⁶ [∵ logₐ a = 1]
6 × 1
Ans. 6

Q.5 log₁₂ 144 + log₁₃ 169 = ?
A. 4
B. 5
C. 6
D. 8

log₁₂ 144 + log₁₃ 169 = ?
log₁₂ (12)² + log₁₃ (13)² = ?
2 + 2
4
Ans. 4

Q.6 is the product of log₁₀ 100 + log₁₀ 1000?
A. 5
B. 10
C. 4
D. 5

log₁₀ 100 + log₁₀ 1000
log₁₀ (10)² + log₁₀ (10)³
2 + 3
5
Ans. 5

Q.7 log₁₀ 125 + log₁₀ 8 = x, if then x = ?
A. 2
B. 3
C. 5
D. 7

log₁₀ 125 + log₁₀ 8 = x,
x = log₁₀ 125 + log₁₀ 8
x = log₁₀ (125 × 8)
x = log₁₀ 1000
x = log₁₀ 10²
x = 2
Ans. 2

Q.8 What is the value of 2log (¹¹⁄₁₃) + 2log (¹³⁰⁄₃₃) − log(⁴⁄₉)?
A. 2.4431
B. 2 log 2
C. log
D. 2 log 5

2log (¹¹⁄₁₃) + 2log (¹³⁰⁄₃₃) − log(⁴⁄₉)
log [(11×11)/(13×13) × (130×130)/(33×33) × (9/4)]
log 25
log 5²
Ans. 2 log 5

Q.9 The simplest value of log₁₀ 25 − 2log₁₀ 3 + log₁₀ 18 is?
A. 18
B. 4
C. log 10³
D. 1

½log₁₀ 25 − 2log₁₀ 3 + log₁₀ 18
log₁₀ 25½ − log₁₀ 3² + log₁₀¹⁸
log₁₀⁵ − log₁₀⁹ + log₁₀¹⁸
log₁₀ (5⁄₉ × 18)
log₁₀¹⁰ = 1
Ans. 1

Q.10 [log₁₀ 50 + log₁₀ 40 + log₁₀ 20 + log₁₀ (2.5)] = ?
A. 3
B. 4
C. 5
D. 10

[log₁₀ 50 + log₁₀ 40 + log₁₀ 20 + log₁₀ (2.5)] = ?
log₁₀ (50 × 40 × 20 × 2.5)
log₁₀ 100000
log₁₀ 10²
5 × 1
Ans. 5

Q.11 What will be the value of log⁷⁵⁄₁₆ − 2log⁵⁄₉ + log³²⁄₂₄₃?
A. 0
B. 1
C. -1
D. log 2

log75⁄₁₆ − 2log⁵⁄₉ + log³²⁄₂₄₃
log75⁄₁₆ − log(⁵⁄₉)² + log³²⁄₂₄₃
log (75 × 81 × 32)/(16 × 25 × 243)
log 2
Ans. log 2

Q.12 If log (2 + 3 + x) = log 2 + log 3 + log x, then x will be?
A. 0
B. 1
C. 2
D. 5

Solution: – According to the question,
log (2 + 3 + x) = log 2 + log 3 + log x
log (5 + x) = log (2 × 3 × x)
5x = 5
x = 1
Ans. 1

Q.13 If 3 log 2 + 2 log 3 + log 5 = log k, then the value of k will be?
A. 360
B. 420
C. 480
D. 524

Solution:- According to the question,
3log 2 + 2 log 3 + log 5 = log k
log 2³ × 3² × 5 = log k
k = 3 × 2 × 5
k = 360

Q.14 If log₁₀ (x² – 6x + 45) = 2, then what is the value of x?
A. 10, 5
B. 11, -5
C. 6, 9
D. 9, -5

हल:- प्रश्नानुसार,
log₁₀ (x² – 6x + 45) = 2
(x² – 6x + 45) = 10²
(x²- 6x + 45) = 100
x² – 6x + 45 – 100 = 0
x² – 6x – 55 = 0
x² – 11x + 5x – 55 = 0
x(x – 11) + 5(x – 11) =0
(x – 11)(x + 5) =0
x – 11 = 0, x = 11
x + 5 = 0, x = -5

Q.15 If log x – 5 log 3 = -2, then the value of x will be?
A. 0.81
B. 1.25
C. 2.43
D. 3.20

Solution:- According to the question,
Log 3⁵ – log 10²
log 243 – log 100
log
x = 2.43
Ans. 2.43

Q.16 What will be the value of log₁₂₉₆ 6?
A. Between 5 and 6
B. 4
C. 0.25
D. 216

Solution:- According to the question,
log₁₂₉₆ 6
log₁₀ 6/log₁₀ 1296
log₁₀ 6/log₁₀ 6⁴
log₁₀ 6/4log₁₀ 6
Ans
. 0.25

Q.17 logₐ 3 = , then the value of a will be?
A. 27
B. 81
C. 27
D. 9

Solution:- As per the question,
logₐ √3 = 1⁄4
a9 = √3
a = (√3)6
a = 33
a = 27
Ans. 27

Q.18 What will be the value of log₄ 8 × 1/log₄ 8?
A. 1
B. 2
C. 0
D. 4

Solution:- According to the question,
log₄ 8 × 1/log₄ 8
Ans. 1

Q.19 If log₁₀ 2 = a and log₁₀ 3 = b then log 5¹² will be equal to?
A. (a + b)/(1 + a)
B. 2a + b/(1 – a)
C. 2a – b/(1 + a)
D. (a – b)/(1 – a)

Solution:- According to the question,
log₅ 12 = log 12/log 5
log 12 /log 10⁄₂
log 12/log 10 – log 2
log (2 × 2 × 3)/1 – log2
2log 2 + log 3/1 – log 2
Ans. 2a + b/(1 – a)

Q.20 log₅ [(125)(625)/25]
A. 25
B. 5
C. 625
D. 5 log₅

Solution:- According to the question,
log₅ (125 × 25)
log (5)⁵
5 log₅
Ans. 5

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